Applied Mathematics and Mechanics was founded in 1980 by CHIEN Wei-zang, a celebrated Chinese scientist in mechanics and mathematics. The current editor in chief is Professor LU Tianjian from Nanjing University of Aeronautics and Astronautics. The Journal was a quarterly in the beginning, a bimonthly the next year, and then a monthly ever since 1985. It carries original research papers on mechanics, mathematical methods in mechanics and interdisciplinary mechanics based on artificial intelligence mathematics. It also strengthens attention to mechanical issues in interdisciplinary fields such as mechanics and information networks, system control, life sciences, ecological sciences, new energy, and new materials, making due contributions to promoting the development of new productive forces.
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Display Method:
2026, Volume 47, Issue 1
publish date：January 01 2026
Display Method:
2026, 47(1): 1-6.
doi: 10.21656/1000-0887.460226
Abstract:
2026, 47(1): 1-14.
doi: 10.21656/1000-0887.460062
Abstract:
To address the challenges of complex modeling and filling of conformal lattices in irregular load-bearing structures, as well as the difficulty in optimization caused by a surge in design variables due to large-scale unit cells, a conformal lattice variable density control design method based on functional descriptions was proposed. The parametric modeling method for conformal lattices based on mesh deformation was developed, enabling rapid lattice filling for irregular structures. Furthermore, a lattice unit cell size control method based on piecewise cubic Hermite interpolating polynomials and a lattice rod diameter control method based on surrogate models were proposed, to achieve fine control of the lattice and dimensionality reduction of design variables. On this basis, an optimization design framework for lattice structures based on an adaptively updated dynamic surrogate model was established, to realize rapid optimization design of lattice control parameters. Two engineering case studies, including the strain energy optimization of rocket payload adapters and the buckling optimization of irregular load-bearing cabin structures for aircraft, validates the proposed method. The comprehensive computational results show the effectiveness of the proposed method for different problems.
To address the challenges of complex modeling and filling of conformal lattices in irregular load-bearing structures, as well as the difficulty in optimization caused by a surge in design variables due to large-scale unit cells, a conformal lattice variable density control design method based on functional descriptions was proposed. The parametric modeling method for conformal lattices based on mesh deformation was developed, enabling rapid lattice filling for irregular structures. Furthermore, a lattice unit cell size control method based on piecewise cubic Hermite interpolating polynomials and a lattice rod diameter control method based on surrogate models were proposed, to achieve fine control of the lattice and dimensionality reduction of design variables. On this basis, an optimization design framework for lattice structures based on an adaptively updated dynamic surrogate model was established, to realize rapid optimization design of lattice control parameters. Two engineering case studies, including the strain energy optimization of rocket payload adapters and the buckling optimization of irregular load-bearing cabin structures for aircraft, validates the proposed method. The comprehensive computational results show the effectiveness of the proposed method for different problems.
2026, 47(1): 15-31.
doi: 10.21656/1000-0887.460132
Abstract:
Peridynamics (PD) is a type of integral nonlocal continuum mechanics, originally developed for analyzing damage, fracture, and failure in solids and structures under complex loadings. It has since been extended to diverse problems including diffusion-transport, fluid mechanics, and multi-physics/chemistry couplings (thermal, electrical, magnetic, hygro, chemical). Recent years have witnessed increasingly diversified dimensions, deepening research focus, and broadening applications within the peridynamics field. Based on the bibliometric statistics and visual text-mining tool CiteSpace, PD’s development history, current research hotspots, and future trends were reviewed. The research evolution of peridynamics can be categorized into the embryonic, the slow-growth, and the fast-growth periods. Leading contributing nations include China, the United States, the United Kingdom, Germany, Italy, and Türkiye. While the US maintains leadership across multiple domains, China exhibits the fastest growth rate, transitioning from following to paralleling and even leading in specific areas. Key research hotspots encompass PD theoretical modeling, numerical algorithms, dynamic crack propagation and severe fracture in solids, elastic-plastic/large deformation problems, multi-physics couplings (notably thermo-mechanical and fluid-solid), multi-scale and homogenization modeling of heterogeneous materials, coupling PD with other methods (e.g., PD-FEM, PD-SPH, PD-MD, PD-FVM), high-performance computing, and the integration of PD with machine learning. These areas are anticipated to remain central to future research of peridynamics.
Peridynamics (PD) is a type of integral nonlocal continuum mechanics, originally developed for analyzing damage, fracture, and failure in solids and structures under complex loadings. It has since been extended to diverse problems including diffusion-transport, fluid mechanics, and multi-physics/chemistry couplings (thermal, electrical, magnetic, hygro, chemical). Recent years have witnessed increasingly diversified dimensions, deepening research focus, and broadening applications within the peridynamics field. Based on the bibliometric statistics and visual text-mining tool CiteSpace, PD’s development history, current research hotspots, and future trends were reviewed. The research evolution of peridynamics can be categorized into the embryonic, the slow-growth, and the fast-growth periods. Leading contributing nations include China, the United States, the United Kingdom, Germany, Italy, and Türkiye. While the US maintains leadership across multiple domains, China exhibits the fastest growth rate, transitioning from following to paralleling and even leading in specific areas. Key research hotspots encompass PD theoretical modeling, numerical algorithms, dynamic crack propagation and severe fracture in solids, elastic-plastic/large deformation problems, multi-physics couplings (notably thermo-mechanical and fluid-solid), multi-scale and homogenization modeling of heterogeneous materials, coupling PD with other methods (e.g., PD-FEM, PD-SPH, PD-MD, PD-FVM), high-performance computing, and the integration of PD with machine learning. These areas are anticipated to remain central to future research of peridynamics.
2026, 47(1): 32-45.
doi: 10.21656/1000-0887.460102
Abstract:
Rapid prediction of mechanical properties of metal cutting is critical to optimal design and productivity improvement of industrial manufacturing. Current prediction models often require expensive and time-consuming experimental and analytical processes. A prediction model based on metal cutting simulation and decision tree regression was constructed to obtain mechanical properties under different cutting conditions. Firstly, the adaptive smoothed particle hydrodynamics (ASPH) was used to simulate the metal cutting process, capture a variety of mechanical properties under different simulation parameters, and form a simulation dataset of 2 000 cutting conditions. Secondly, the decision tree regression (DTR) was used to learn the simulation data set, train and construct the metal cutting prediction model, and evaluate the effect of the prediction model under different pruning strategies by cross-validation and grid search. The results show that, the established prediction model can quickly predict multi-mechanical properties under different simulation parameters, and the appropriate pruning strategy can improve the accuracy, generalization ability and stability of the prediction model.
Rapid prediction of mechanical properties of metal cutting is critical to optimal design and productivity improvement of industrial manufacturing. Current prediction models often require expensive and time-consuming experimental and analytical processes. A prediction model based on metal cutting simulation and decision tree regression was constructed to obtain mechanical properties under different cutting conditions. Firstly, the adaptive smoothed particle hydrodynamics (ASPH) was used to simulate the metal cutting process, capture a variety of mechanical properties under different simulation parameters, and form a simulation dataset of 2 000 cutting conditions. Secondly, the decision tree regression (DTR) was used to learn the simulation data set, train and construct the metal cutting prediction model, and evaluate the effect of the prediction model under different pruning strategies by cross-validation and grid search. The results show that, the established prediction model can quickly predict multi-mechanical properties under different simulation parameters, and the appropriate pruning strategy can improve the accuracy, generalization ability and stability of the prediction model.
2026, 47(1): 46-56.
doi: 10.21656/1000-0887.450327
Abstract:
Based on the 1st-order shear deformation theory (FSDT) and the potential flow theory, the transverse vibration of functionally graded material (FGM) cylinder bars dipped in fluid was analyzed. The cermet bar material properties following a power-law distribution along the radial direction were represented by the radial gradient index. The fluid velocity potential and hydrodynamic loads were determined through solution of the Laplace equation in cylindrical coordinates with the variable separation method. The governing equations of motion were derived according to Hamilton’s principle. The fundamental frequencies and mode shapes were obtained with the generalized differential quadrature (GDQ) method and the direct iteration method. Additionally, the finite element analysis (CEL simulation) was used to validate the numerical results. Through parametric studies, the effects of the length-to-diameter ratio, the gradient index, the boundary conditions, as well as the fluid depth and density, on the transverse vibration behavior of the FGM bar-fluid interaction system were evaluated.
Based on the 1st-order shear deformation theory (FSDT) and the potential flow theory, the transverse vibration of functionally graded material (FGM) cylinder bars dipped in fluid was analyzed. The cermet bar material properties following a power-law distribution along the radial direction were represented by the radial gradient index. The fluid velocity potential and hydrodynamic loads were determined through solution of the Laplace equation in cylindrical coordinates with the variable separation method. The governing equations of motion were derived according to Hamilton’s principle. The fundamental frequencies and mode shapes were obtained with the generalized differential quadrature (GDQ) method and the direct iteration method. Additionally, the finite element analysis (CEL simulation) was used to validate the numerical results. Through parametric studies, the effects of the length-to-diameter ratio, the gradient index, the boundary conditions, as well as the fluid depth and density, on the transverse vibration behavior of the FGM bar-fluid interaction system were evaluated.
2026, 47(1): 57-67.
doi: 10.21656/1000-0887.460044
Abstract:
Through quantitative characterization of the effects of various strengthening mechanisms on the yield strengths of nanoparticlereinforced metal matrix composites (NRMMCs) within a wide temperature range, as well as the impacts of grain boundary sliding on the yield strength of the metal matrix, a theoretical characterization model for the temperaturedependent yield strengths of NRMMCs without fitting parameters was established. This model only requires the yield strength of the metal matrix at any one reference temperature and relevant material parameters such as the specific heat capacity, the thermal expansion coefficient, and the melting point, etc., to predict the yield strengths of the NRMMCs at any temperature. The predicted results of the model are in good agreement with all the 4 sets of experimental data currently available, achieving a reasonable prediction of the yield strengths of the NRMMCs within a wide temperature range. On this basis, the effects of the main strengthening mechanisms on the yield strengths of the NRMMCs and their evolution laws with the temperature and the particle size were discussed, to provide a theoretical basis and effective suggestions for the design and development of NRMMCs applicable to a wide temperature range.
Through quantitative characterization of the effects of various strengthening mechanisms on the yield strengths of nanoparticlereinforced metal matrix composites (NRMMCs) within a wide temperature range, as well as the impacts of grain boundary sliding on the yield strength of the metal matrix, a theoretical characterization model for the temperaturedependent yield strengths of NRMMCs without fitting parameters was established. This model only requires the yield strength of the metal matrix at any one reference temperature and relevant material parameters such as the specific heat capacity, the thermal expansion coefficient, and the melting point, etc., to predict the yield strengths of the NRMMCs at any temperature. The predicted results of the model are in good agreement with all the 4 sets of experimental data currently available, achieving a reasonable prediction of the yield strengths of the NRMMCs within a wide temperature range. On this basis, the effects of the main strengthening mechanisms on the yield strengths of the NRMMCs and their evolution laws with the temperature and the particle size were discussed, to provide a theoretical basis and effective suggestions for the design and development of NRMMCs applicable to a wide temperature range.
2026, 47(1): 68-78.
doi: 10.21656/1000-0887.460057
Abstract:
The longitudinal critically refracted (LCR) wave method demonstrates significant advantages in non-destructive testing of structural stresses. However, the current LCR wave technique lacks a physically meaningful analytical relationship between effective detection depths and excitation frequencies of incident waves, and cannot be applied to measure non-uniform stress distributions along the component thickness. To address these limitations, an analytical expression for the correlation between effective detection depths and excitation frequencies in the LCR wave method-based stress measurements, was derived. A stepwise difference algorithm was proposed for characterizing non-uniform stress fields with the LCR wave method, and through numerical simulations to validate its effectiveness in measuring inhomogeneous stress distributions.
2026, 47(1): 79-89.
doi: 10.21656/1000-0887.450308
Abstract:
The 2D plane problem of non-circular nanoholes under uniform far-field heat flow was investigated. To examine the effects of surface phonon scattering on heat conduction at the microscale, a weakly thermal conductivity model considering temperature jump was introduced, and the complete Gurtin-Murdoch lower-order surface energy model was employed to characterize the surface effects. Based on the theory of complex functions and series expansion, the series solutions for the temperature field and the thermal stress field were obtained corresponding to different hole shapes defined with the conformal mapping techniques. Several numerical examples of non-circular nanoholes were presented to analyze the surface effects on thermal stress fields. The results indicate that, surface effects significantly increase the thermal stresses around the nanohole, and the combined action of surface elasticity and surface tension plays a key role in determining the magnitude of the thermal stress.
The 2D plane problem of non-circular nanoholes under uniform far-field heat flow was investigated. To examine the effects of surface phonon scattering on heat conduction at the microscale, a weakly thermal conductivity model considering temperature jump was introduced, and the complete Gurtin-Murdoch lower-order surface energy model was employed to characterize the surface effects. Based on the theory of complex functions and series expansion, the series solutions for the temperature field and the thermal stress field were obtained corresponding to different hole shapes defined with the conformal mapping techniques. Several numerical examples of non-circular nanoholes were presented to analyze the surface effects on thermal stress fields. The results indicate that, surface effects significantly increase the thermal stresses around the nanohole, and the combined action of surface elasticity and surface tension plays a key role in determining the magnitude of the thermal stress.
2026, 47(1): 90-100.
doi: 10.21656/1000-0887.460002
Abstract:
The Holling-Ⅱ functional responses and an improved Leslie-Gower term were considered to establish a cross-diffusion predator-prey model with double Allee effects. The existence and stability of positive equilibrium points were analyzed in the absence of diffusion to provide conditions for Turing instability under the diffusion effects. The influential mechanisms of the double Allee effects on the pattern formation, the structural changes, and the evolutionary speed was mainly investigated. The findings reveal that, in stable diffusion-driven systems, the Allee effects can induce pattern formation; conversely, in unstable systems, the Allee effects can lead to structural changes in patterns. Additionally, the time required for the system to reach stable homogeneous and mixed patterns varies with different Allee effects coefficients, indicating that the Allee effects can significantly alter the evolutionary speed of patterns. Therefore, the double Allee effects plays a crucial role in the formation and evolution of Turing patterns in predator-prey systems.
The Holling-Ⅱ functional responses and an improved Leslie-Gower term were considered to establish a cross-diffusion predator-prey model with double Allee effects. The existence and stability of positive equilibrium points were analyzed in the absence of diffusion to provide conditions for Turing instability under the diffusion effects. The influential mechanisms of the double Allee effects on the pattern formation, the structural changes, and the evolutionary speed was mainly investigated. The findings reveal that, in stable diffusion-driven systems, the Allee effects can induce pattern formation; conversely, in unstable systems, the Allee effects can lead to structural changes in patterns. Additionally, the time required for the system to reach stable homogeneous and mixed patterns varies with different Allee effects coefficients, indicating that the Allee effects can significantly alter the evolutionary speed of patterns. Therefore, the double Allee effects plays a crucial role in the formation and evolution of Turing patterns in predator-prey systems.
2026, 47(1): 101-112.
doi: 10.21656/1000-0887.460008
Abstract:
The global exponential synchronization of a class of memristive neural networks with proportional delays and uncertain parameters was studied. Firstly, the error system of the drive-response systems was established. Secondly, the exponential function was introduced and an adaptive controller was designed, to divide the error system into 4 cases according to the switching characteristics of the system, then the proper Lyapunov functional was constructed and combined with the mean value inequality, and the global exponential synchronization criterion for the studied system was obtained. Meanwhile, the case of global exponential synchronization degenerating into global asymptotic synchronization was considered. Finally, the effectiveness of the obtained criterion was verified through a numerical example and simulation.
The global exponential synchronization of a class of memristive neural networks with proportional delays and uncertain parameters was studied. Firstly, the error system of the drive-response systems was established. Secondly, the exponential function was introduced and an adaptive controller was designed, to divide the error system into 4 cases according to the switching characteristics of the system, then the proper Lyapunov functional was constructed and combined with the mean value inequality, and the global exponential synchronization criterion for the studied system was obtained. Meanwhile, the case of global exponential synchronization degenerating into global asymptotic synchronization was considered. Finally, the effectiveness of the obtained criterion was verified through a numerical example and simulation.
2026, 47(1): 113-122.
doi: 10.21656/1000-0887.460090
Abstract:
An energy-preserving numerical algorithm, which is 2nd-order in time and 4th-order in space, for nonlinear wave equations was developed based on the order reduction method, the Lagrange multiplier method and the compact difference method. The discrete original energy conservation property of the suggested algorithm was proven. The computational procedure of the associated algorithm was exhibited. Numerical results validate the exactness and effectiveness of the proposed algorithm.
An energy-preserving numerical algorithm, which is 2nd-order in time and 4th-order in space, for nonlinear wave equations was developed based on the order reduction method, the Lagrange multiplier method and the compact difference method. The discrete original energy conservation property of the suggested algorithm was proven. The computational procedure of the associated algorithm was exhibited. Numerical results validate the exactness and effectiveness of the proposed algorithm.
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Founded in 1980 ( monthly )
Supervisor：Chongqing Jiaotong University
Founder：CHIEN Weizang
Editor-in-Chief：LU Tianjian
ISSN 1000-0887
CN 50-1060/O3
